Data Representation – Chapter 3 Sections 3-2, 3-3, 3-4.

Data Representation – Chapter 3

Sections 3-2, 3-3, 3-4

Homework Assignment

• Questions/Requests/Comments/Concerns?

Number Range

• What is meant by “range”?

• How do you compute the range of a given set of digits?

More Representations

• Integer values are represented by a power series regardless of radix (base)1410 = 1 x 101 + 4 x 100

11102 = 1 x 23 + 1 x 22 + 1 x 21 + 0 x 20

• What about fractions?

Fraction Representations• Same principle applies

14.3710 = 1 x 101 + 4 x 100 + 3 x 10-1 + 7 x 10-2

“decimal point”

10.112 = 1 x 21 + 0 x 20 + 1 x 2-1 + 1 x 2-2

“binary point”

Fraction Representations• To convert a decimal fraction to binary

– Multiple decimal fractional part by 2– Output the integer part– Repeat until you’ve reached the desired accuracy

0.312510 = 0.01012

.3125 x 20.6250

.6250 x 21.2500

.2500 x 20.5000

.5000 x 21.0000

Fraction Representations• More on binary floating point

representations later (maybe)

Addition

• Decimal number addition

carry121

+19

40

Addition

• Binary number addition

1 111 carries

10101

+10011

101000• It works the same way in binary as it

does in decimal

Subtraction

• Decimal number subtraction

1 1 borrow

21

+19

02

Subtraction

• Binary number subtraction

01 borrow

10101

-10011

00010• It works the same way in binary as it

does in decimal

Subtraction Questions

• What about when a subtraction results in a negative value?

• How do you represent a negative value in binary?

• How do you represent a negative value in decimal?

Negative Numbers

• Decimal

- 27

• Place a “-” in front of the decimal digits

Negative Numbers

• Binary

-11011

• Place a “-” in front of the binary digits

• This is called signed-magnitude representation

• In the hardware the sign is a designated bit where 0 means “+” and 1 mean “–”

Negative Numbers

• Signed-magnitude is convenient for humans doing symbolic manipulations

• It’s not convenient for computer computations

• Why not?– Consider subtraction (which is really

addition of a negative number)

Signed-Magnitude Subtraction

21 Minuend

-17 Subtrahend

4 Difference

17 Minuend

-21 Subtrahend

- 4 Difference

Signed-Magnitude Subtraction

if (M >= S)compute M-S

elsecompute S-Mplace “-” on difference

• To perform subtraction we must first perform a comparison!– Therein lies the inconvenience

Negative Numbers Revisited• Complement representation

• “(r-1)’s” complement– Given Nr , an n-digit number of radix r

– “(r-1)’s” complement is (rn – 1) - Nr

“(r-1)’s” complement

• Decimal1234510 n=5, r=10, N=12345

(105-1)-12345

(100000-1)-12345

99999-12345

87654– “9’s” complement


• Binary101102 n=5, r=2, N=10110(25-1)-10110(100000-1)-1011011111-1011001001

– “1’s” complement– Note that the result is just an inversion of

the bits of the original number (1/0 and 0/1)


• So, how does this help us do subtraction?– It doesn’t really!– Furthermore the representation of 0 is

ambiguous

0000 “+0”

1111 “-0”

Try it!

Another Complement

• “r’s” complement– Given Nr , an n-digit number of radix r

– “r’s” complement is “(r-1)’s” complement + 1

(rn – 1) – Nr+ 1 rn – Nr if (Nr != 0)

0 if (Nr == 0)

“r’s” complement

• Let’s concentrate on the 2’s complement (binary number system)– “…leaving all least significant 0’s and the

first 1 unchanged, and then replacing 1’s by 0’s and 0’s by 1’s in all other higher significant bits.”

• Yeah, right!– Take the “1’s” complement and add 1

“2’s” Complement

101102 n=5, r=2, N=10110

(25-1)-10110

(100000-1)-10110

11111-10110

01001 1’s complement

+ 1

01010 2’s complement

2’s Complement “Subtraction”

0111

-0101 0111

+1011

10010

Original Problem 2’s Complement Problem

Carry Out Answer2’s Complement


0101

-0111 0101

+1001

01110

Original Problem 2’s Complement Problem



0101

+1001

01110

2’s Complement Problem


0111

+1011

10010



• How do we know the sign of the result?

2’s Complement “Subtraction”• Recall that we’re doing “fixed bit-length”

math

• When the numbers are “signed” then the most significant bit (MSB) represents the sign– MSB=1 defined as negative– MSB=0 defined as positive

Sign of the Result?

0101

+1001

01110



0111

+1011

10010



Sign Bit Sign Bit

Sign of the Result?

• The result is in 2’s complement form– If the sign bit is 1, take the 2’s complement

of the result to read off the magnitude in “normal” binary notation

– If the sign bit is 0, the result is in “normal” binary notation

Overflow

• What if the result is too big?– Remember, this is fixed bit-length math– When you add to n-bit numbers the result

may be up to n+1 bits long

Overflow

• When a signed positive number is added to a signed (2’s complement) negative number an overflow cannot occur– The positive number magnitude will only get smaller

• When two signed negative numbers or two positive numbers are added an overflow may occur– Magnitude is going to get bigger

• When two signed positive numbers are added, the sign bit is treated as part of the magnitude of the number and the end carry (overflow) is not treated as “overflow”

Overflow

• So, when is that “extra bit” an overflow and when is it part of the result?– Check the sign bit and the overflow bit– If the two are not equal then an overflow

has occurred– If the two are equal then there is no

overflow

Overflow

• In an overflow condition– If the numbers are both positive, then the

sign bit becomes part of the magnitude of the result

– If the numbers are both negative, then the overflow bit is treated as the sign

• As we’ll see next time (logic gates) there is a very simple way to detect overflow

• Questions?

“2’s” Complement

• Is this cheating on the representation of 0 since we handled it with an conditional statement?

• No! Try it based on the formula only!

• Don’t really need to specify it as two cases

• I only did it because the book did

“2’s” Complement• So, how does this help us do subtraction?• Addition of a 2’s complement subtrahend.

– Take 2’s complement of the subtrahend (negate the subtrahend)

– Add minuend to subtrahend– Difference is in 2’s complement notation

• Note we’re using a fixed, pre-specified number of bits

• If the result overflows, just ignore it for now

Other Binary Formats• Binary Coded Decimal (BCD)

– Convert each decimal digit to its binary representation and store– Requires 4 bits per digit – why?– Forget about BCD arithmetic

• Even parity/Odd parity– For a binary number, count the number of 1’s– If the number is odd, attach an extra “1” for even parity or a “0” for odd parity– If the number is even, attach an extra “0” for even parity or a “1” for odd parity– What good is this?

• Gray code– Count through a sequence of binary numbers in such a way that only 1 bit at a time

changes– What good is this?

• American Standard Code for Information Interchange (ASCII)– Standard set of binary patterns assigned to 256 characters– How many bits comprise the ASCII code?

• Unicode– Up-to-date replacement for ASCII– Various schemes utilizing various bit lengths

Homework

• Textbook pages 89 – 91– 3-9, 3-10, 3-13, 3-15, 3-16, 3-17, 3-23

– Due beginning of next lecture

• Begin reading Chapter 1

Data Representation – Chapter 3 Sections 3-2, 3-3, 3-4.

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